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Posted by on Apr 7, 2007 in mirror puzzle | 18 comments

Mirror puzzle – solution?

What follows is my own suggested solution to this puzzle – scroll down for the puzzle (I note several of you offer much the same solution)

We noted before that, in a sense, mirrors don’t reverse anything. So why do we say they reverse left-to-right, but not top-to-bottom? Well, if the mirror before you was replaced by a sheet of glass, and you were to stand behind the glass in just the position your mirror-self seems to stand, then while your head would still be at the top and your feet at the bottom, your left hand would be over to the right, where your right hand appears in the mirror, and your left hand would be to the left, where your left hand appears. That is why we say the mirror reverses left-to-right, but not top-to-bottom.

But notice that we have just taken something for granted: the axis about which we rotate you when we imagine you over there behind the mirror. When we turn something round, we rotate it about an axis. A spinning top, for example, rotates around a vertical axis. A car wheel rotates around a horizontal axis. When we imagine you over there in the position your mirror-self seems to be in, we mentally put you there by rotating you about a vertical axis. But what if we were to get you over there not by rotating you around a vertical axis, but a round a horizontal axis? Then you would be stood on your head. And, compared to your mirror image, your left and right sides would not then be switched round. Your left hand (the one with the watch in the diagram) remains to the left. Which is also where your left hand appears in the reflection. But top and bottom are now reversed. Your head appears where your feet are in the image.

It seems the reason we say mirrors reverse left and right but not top and bottom is due to the fact that we take for granted a particular axis of rotation. But we could just as easily choose a horizontal axis. Then it would be true to say that a mirror reverses top to bottom but not left to right.

So yes, it is true to say mirrors reverse left to right, but only if we choose a vertical axis of rotation. Choose a horizontal axis and they then reverse top to bottom.

Of course, this raises the question of why we take the vertical axis for granted. The answer, presumably, is that people are not in the habit of flying through the air like birds and settling on their heads. When people normally rotate, it is almost always about a vertical axis. So we just took for granted a vertical axis of rotation in this case too.

So this puzzle about why mirrors do what they do is generated by our not noticing what has been taken for granted. To solve the puzzle, we need to take a step back and start questioning what we took for granted.

When only philosophy will do

Notice that, if this solution (or part solution) is correct, we certainly didn’t have to any scientific research into how light and mirrors behave. Nor did we have to investigate how our brains work. Even if we had done that sort of scientific research, it still wouldn’t have solved the puzzle. In order to solve this puzzle, we need to stop doing science and start doing philosophy. It is a puzzle that is solved just by thinking.

People sometimes assume all questions can be answered by science. They would assume that the mirror puzzle must have a scientific solution. But it turns out that the mirror puzzle is a puzzle that science cannot solve. It seems that, sometimes, only philosophy will do.

The Door Puzzle

There is similar puzzle about doors. Walk though a door that opens on your left and turn round to come back through it, and the door now opens on your right. But pass through a door that opens at the top (like a cat-flap) and turn to come back through it and the door still opens at the top. Why does passing through a door reverse the way it opens from left to right, but not from top to bottom? What explains the difference?

The solution is much the same as for the mirror puzzle. When you pass through a left-opening door and turn around to come back through it, you would normally rotate about a vertical axis. But what if you were to rotate about a horizontal axis, and you floated back through upside down? Then the door that opened on the left would still open the left on the way back though it, but a door that opened at the bottom would now open at the top. We say that left and right are reversed but not top and bottom only because we take for granted a particular axis of rotation.

In space, where we are weightless, the axis of rotation about which we choose to rotate when turning to come back through a door is less likely to be the vertical axis. You could just as easily spin about a horizontal axis instead. So, after years in space, it might seem as natural to you to say that a door that opens at the top opens at the bottom when you come back through it as it does to say that a door that opens on the left opens on the right when you return though it.

For creatures that live in a weightless environment, where it is as easy to rotate about one axis as the other, perhaps neither the mirror puzzle nor the door puzzle would even be puzzles.


  1. Interestingly enough this question was put to me by my high school (11th grade) physics teacher. It took me a couple of years to work it out, and another couple of years to be confident of the answer.But this question seems—contrary to your earlier post—to be very atypical of the sorts of problems raised in the philosophical canon, in that it has an actual solution.

  2. “For creatures that live in a weightless environment, where it is as easy to rotate about one axis as the other, perhaps neither the mirror puzzle nor the door puzzle would even be puzzles.”This last sentence is the key for me. Most humans have a poor intuition for certain concepts in mathematics (probability being the most obvious example).A reflection is not the same transform as a rotation, and I think because the human psychology deals with rotation more it presupposes rotation. (i.e. when facing another human they are rotated, so your mental rule of thumb is rotation).

  3. SL said: “People sometimes assume all questions can be answered by science. They would assume that the mirror puzzle must have a scientific solution. But it turns out that the mirror puzzle is a puzzle that science cannot solve. It seems that, sometimes, only philosophy will do.”I’m only a neophyte when it comes to philosophy and science, and I do not “assume all questions can be answered by science,” but the explanation of what is happening in the Mirror Puzzle seems to me more scientific than philosophical: the affect of reflected electromagnetic waves on the cognitive behavior of a flightless land creature which evolved in a gravitational field (i.e., vertical axis of rotation assumption). It’s mostly about physics and biology (philosophy of science and philosophy of mind figure in as well, I suppose). Your similar Door Puzzle literally adds rocket science.SL: “It is a puzzle that is solved just by thinking.”True, compared to the expense of scientific research, these two thought experiments are a bit of “no money fun,” but I don’t think we would have these particular puzzles, and their solutions, had that “money” not been spent by scientists who came before us. If we had been discussing the Mirror Puzzle at the feet of Aristotle, for example, the answer might have included something about perfect solids and the ether. In other words, without science to inform us of matters of fact, philosophical inquiry alone can easily lead us in the wrong direction.

  4. What I had in mind when I said it was not a puzzle science can answer is – I included all the relevant scientific facts when I set the puzzle up, and they didn’t answer it.The answer is produced by reflecting on the concepts involved (top, bottom, left and right) and what we take for granted when applying them.So it was a sort of conceptual clarification that was required. Not more scientific theory. And conceptual clarification is, I take it, one of the core activities of philosophy.I do take the point that this is an atypical philosophical puzzle in that it does have a clear-ish solution (though there are papers published which suggest it actually goes much deeper than we have supposed – including a classic paper by Lycan).

  5. In fairness, I think you set up the puzzle in a slightly misleading manner. It wasn’t clear, to me at least, that this was purely about our perception of object orientation. Superficially, it seemed like a straightforward question about how a mirror works, which clearly caught a few others out as well, in so much that physics alone is sufficient to explain this phenomenon. Or maybe you did set it up out correctly, and we’re just victims of scientific conditioning. That’s probably a legitimate rebuttal right there.Here’s a question though: it’s taken for granted that the visual cortex ‘corrects’ the inverted image projected onto the retina before sending it out to the higher areas of the cortex etc. Is such a correction really needed? Can we even ascribe spatial dimensions to what amounts to little more than a complex synchronised firing of neurones? I think this relates to what Pinker (I think it was him, or maybe he was quoting somebody else) termed the “the big question”, which for now sets it beyond the realm of empirical science, but might be more suitable for a philosopher to ponder.

  6. Yes, people take direction for granted. If exposed to it enough, people take any kind of information for granted.Like clothes: you’ve been wearing them for so long, I bet you don’t even feel it. Or like habits: you can walk without thinking about it.Once information is taken for granted, or well-known, by everyone, it becomes ‘common sense,’ even if people don’t know the reason for the information.And of course, ‘common sense’ can be completely out of wack. (Seriously, a flat Earth in the center of the Universe?)Everyone’s seen a mirror, then everyone takes for granted mirrors reverse images vertically.But of course, not everyone’s seen a CONCAVE mirror. There’s where ‘common sense’ fails. In a concave mirror, you can see yourself upside-down, with left and right in their proper place.People who can let go of common sense can be philosophers.P.S: In addition to Rev. Dr. Incitatus, our brain has this habit of trying to ‘correct’ or ‘make sense’ of things. (That’s why we have philosophers, religion, and science) Of course, how we ‘correct’ something is based on previous knowledge. We all know (some) mirrors reflect left as right. We also know when someone puts out his/her left hand out in front of us, it looks like it’s on the right. We also know to be in front of someone, you have to ROTATE VERTICALLY 180 DEGREES. Most of us would then string all this information and come up with a conclusion that’s wrong yet it becomes common sense ANYWAY: The ‘most of us’ would infer that mirrors ROTATE my image.People who are not ‘most of us’ would give this matter more thought and research before surmising a conclusion.Inferring without giving stuff enough thought is a dangerous thing to do.

  7. “our brain has this habit of trying to ‘correct’ or ‘make sense’ of things. (That’s why we have philosophers, religion, and science) Of course, how we ‘correct’ something is based on previous knowledge.”Cargo cults being a fine example of this.

  8. *BRAINSURGE*When I read this puzzle, it left me confused and pondering for long periods of times, even at school or during meals.The question Stephen Law raised “Why do mirrors reflect left as right instead of up as down?” is deceptional. A trick question.Up stays up.Left actually stays left too.There are 3 dimensions in this world: left-right, up-down, and front-back.When you face a mirror directly, the dimensions of left-right and up-down stay the same when reflected because they are parellel to the mirror.Front-back, however, isn’t.The dimension of front-back is perpendicular to the mirror, reversing everything to back-front in the mirror.I wish I could make it clearer with a thought experiment. But bottom line, left-right isn’t reversed, front-back is.”So WHY does my mirror image raise his left hand when I raise my right?”Our perception of left-right is dependent on front-back. By the way, front-back is also dependent on up-down. Why? That’s the way humans ‘common sensified’ direction. Foolish humans. >:)

  9. The perpendicular mirror reflects, point to point, end of story. The reflection you see isn’t real, so when you raise your left hand you’re seeing the reflection of your left hand rise, it is not the image-you’s right hand. If you are wearing a ring on your left hand but not your right, which hand is being raised? The ring hand. The terms ‘left hand’ ‘right hand’ are being misused. A completely spherical being wouldn’t have this problem.The clock example is misrepresented too. It does have the same apparent reflection problem, in that its 3 is now where its 9 should be.This is not a philosophical puzzle. It’s not even a science puzzle. It’s not a puzzle! Philosophy is not required, though thinking is, and that seems to be surprisingly absent. It appears that the real problem is that with only one line of symmetry many people find it difficult to know their arse from their elbow.

  10. Quote: “People sometimes assume all questions can be answered by science. They would assume that the mirror puzzle must have a scientific solution. But it turns out that the mirror puzzle is a puzzle that science cannot solve. It seems that, sometimes, only philosophy will do.”Stephen, correct me if I’m wrong, but you seem to argue that deductive reasoning is entirely the purview of philosophy and that inductive reasoning and empirical observation are entirely the purview of science. This cannot be farther from the truth.Yes, science rests upon the foundation of empirical observation, but these empirical observations are typically codified into concepts and a precise language that allow for direct manipulation of terms using deductive reasoning. What is mathematics but precisely the application of deductive reasoning? What you are essentially arguing in your piece is simply the application and exposition of logical arguments – how is that different from science?Lastly, neither I nor the scientists that I have come across assume that science can solve all problems. It simply looks at all observable phenomena. Science cannot answer metaphysical questions, it can clarify but not answer many epistemological issues, and it is not going to provide you with any normative ethical systems. Philosophy still has its place.

  11. Hi PaulThanks for the link. I think your solution is essentially the same, it’s just a matter of how we describe what the mirror does – e.g. reverses nothing ( which is how I put it) or reverses back to front (which is how you put it). I suspect that “reverses nothing” isn’t entirely problem free, but is probably preferable to “reverses back to front” as the puzzle arises even for objects that lack depth. However, maybe by “back to front” you mean, look from rear as normally look from front? But then that’s not right either – a sheet of paper from rear looks different to how it normally looks from front, because you cannot see the writing on the front (which is why you then start talking about transparent sheets of paper). I think my way of describing situation is simpler, if not perfect. Ultimately, my solution doesn’t require we describe the mirror as reversing anything. We can just draw the diagram of what happens optically, and then explain why we say it reverses in one way not the other in just the way I do.

  12. Hi Stephen,Thanks for your response.One of my points of contention is that this is not a philosophical question but a scientific one, and as one of your contributors, The Barefoot Bum, points out, it does have ‘an actual solution’.Another of your contributors, Nutcasenightmare, in his ‘BRAINSURGE’ comment, then give us that solution quite succinctly and accurately. This is a physics problem, pure and simple.Thanks, Paul.

  13. Hi PaulBut providing someone with all the info about what happens to light etc does not solve the puzzle.So the solution is not scientific in that sense. That was the point I was making.The solution requires thinking about what is taken for granted by describing something as “reversed left right but not top bottom”. It is more of a conceptual/meaning solution, I’d suggest, that involves unpacking what we *mean* by describing something as “reversed”. We are doing what Simon Blackburn calls “conceptual engineering”.

  14. Thanks Stephen for the interaction.I will create a link from my blog to yours (regarding this topic), assuming that’s okay with you.Best regards, Paul.

  15. Sure, Paul. Thanks for contributing. We don’t agree, but hey, that’s what makes this interesting!

  16. “So yes, it is true to say mirrors reverse left to right, but only if we choose a vertical axis of rotation. Choose a horizontal axis and they then reverse top to bottom.”I am probably getting fruitlessly sidetracked, but what could it mean to “choose” an axis here? It rather seems that my experience when looking at my mirror-image is necessary, and not any kind of matter of choice at all. Is there something I can do to see my reflection differently?It is undoubtedly true that, in claiming that mirrors reverse left-right, we are assuming that the vertical axis is the axis in reference to which the judgment is being made. But this is just to explicate the content of the original judgment, and not to explain the state of the world in virtue of which the judgment is appropriate. And, again, I am probably borrowing more trouble than I need, but I would like to think that there is something about the real state of my mirror-image which grounds my judgment of left-right reversal, and my judgment that the vertical axis is *the* axis of concern.Sorry if that is as clear as mud. All that to say — mirrors still puzzle me.

  17. I hope this is a simple answer to the question:Put on a white t-shirt and position yourself facing north. Write in front of your t-shirt north, and on the back south. On your left side, write west, on your right side write east. On top of the t-shirt write up, at the bottom write down.Place a mirror in front of you. You’ll see that your mirror image (m.i.) still has a correct east, west, up and down marker. However, your m.i. faces south but m.i.’s t-shirt says north.Place the mirror left or right of you. You’ll see that m.i.’s t-shirt has a correct north, south, up and down marker. However, your m.i. t-shirt’s west is in the east.Place the mirror above or below yourself. You’ll see that m.i.’s t-shirt has a correct north, south, west and east marker. Only you’re seeing yourself upside down.In all cases, the mirror only mirrors one axis. And, in all cases, when you lift your right hand, your m.i.’s left hand will be risen.How is THAT possible? simple: we defined left and right relative to ourselves: standing upright, facing north we defined ‘left’ as the arm in the west. But turn yourself over any axis by 180° and your left arm will point to the east. What a mirror does is actually not TURN, but INVERT one axis (and it can be upside down, so the philosophical puzzle was flawed to start with), which has the same effect.So, the answer is: up and down are relative to gravity, so up and down are only inverted for horizontal mirrors. Left and right are relative to our position, so are inverted always by a mirror. Any letters on our t-shirt were always written in mirror writ for the samereason.What does it learn me? That we, humans, have it hard to see beyond our own definitions once we consider them ‘natural’.

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